Implementing Automatic Benders Decomposition in a Modern MIP Solver

We describe the automatic Benders decomposition implemented in the commercial solver IBM CPLEX. We propose several improvements to the state-of-the-art along two lines: making a numerically robust method able to deal with the general case and improving the efficiency of the method on models amenable to decomposition. For the former, we deal with: unboundedness, failures in generating cuts and scaling of the artificial variable representing the objective. For the latter, we propose a new technique to handle so-called generalized bound constraints and we use different types of normalization conditions in the Cut Generating LPs. We present computational experiments aimed at assessing the importance of the various enhancements. In particular, on our test bed of models amenable to a decomposition, our implementation is approximately 5 times faster than CPLEX default branch-and-cut. A remarkable result is that, on the same test bed, default branch-and-cut is faster than a Benders decomposition that doesn’t implement our improvements.